Theorem ttcwf2 36707

Description: If a transitive closure class is a set, then it is well-founded, assuming Regularity. (Contributed by Matthew House, 6-Apr-2026.)

Assertion

Ref	Expression
ttcwf2	⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Proof of Theorem ttcwf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)))
2	1	eldifad 3901	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ TC+ 𝐴)
3		ttctr2 36676	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ TC+ 𝐴 → 𝑥 ⊆ TC+ 𝐴)
4	2, 3	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ TC+ 𝐴)
5		dfss2 3907	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ TC+ 𝐴 ↔ (𝑥 ∩ TC+ 𝐴) = 𝑥)
6	4, 5	sylib 218	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) = 𝑥)
7		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
8		inssdif0 4314	. . . . . . . . . . . 12 ⊢ ((𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On) ↔ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
9	7, 8	sylibr 234	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → (𝑥 ∩ TC+ 𝐴) ⊆ ∪ (𝑅1 “ On))
10	6, 9	eqsstrrd 3957	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ⊆ ∪ (𝑅1 “ On))
11		vex 3433	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12	11	r1elss 9730	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On))
13	10, 12	sylibr 234	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → 𝑥 ∈ ∪ (𝑅1 “ On))
14	1	eldifbd 3902	. . . . . . . . 9 ⊢ ((𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∧ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅) → ¬ 𝑥 ∈ ∪ (𝑅1 “ On))
15	13, 14	pm2.65da 817	. . . . . . . 8 ⊢ (𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) → ¬ (𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
16	15	nrex 3065	. . . . . . 7 ⊢ ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅
17	16	a1i 11	. . . . . 6 ⊢ (TC+ 𝐴 ∈ V → ¬ ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
18		difexg 5270	. . . . . . 7 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V)
19		zfreg 9511	. . . . . . 7 ⊢ (((TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
20	18, 19	sylan 581	. . . . . 6 ⊢ ((TC+ 𝐴 ∈ V ∧ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅) → ∃𝑥 ∈ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))(𝑥 ∩ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On))) = ∅)
21	17, 20	mtand 816	. . . . 5 ⊢ (TC+ 𝐴 ∈ V → ¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅)
22		nne 2936	. . . . 5 ⊢ (¬ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) ≠ ∅ ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
23	21, 22	sylib 218	. . . 4 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
24		ssdif0 4306	. . . 4 ⊢ (TC+ 𝐴 ⊆ ∪ (𝑅1 “ On) ↔ (TC+ 𝐴 ∖ ∪ (𝑅1 “ On)) = ∅)
25	23, 24	sylibr 234	. . 3 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))
26		eleq1 2824	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On)))
27		sseq1 3947	. . . . 5 ⊢ (𝑥 = TC+ 𝐴 → (𝑥 ⊆ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
28	26, 27	bibi12d 345	. . . 4 ⊢ (𝑥 = TC+ 𝐴 → ((𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) ↔ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On))))
29	28, 12	vtoclg 3499	. . 3 ⊢ (TC+ 𝐴 ∈ V → (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) ↔ TC+ 𝐴 ⊆ ∪ (𝑅1 “ On)))
30	25, 29	mpbird 257	. 2 ⊢ (TC+ 𝐴 ∈ V → TC+ 𝐴 ∈ ∪ (𝑅1 “ On))
31		elex 3450	. 2 ⊢ (TC+ 𝐴 ∈ ∪ (𝑅1 “ On) → TC+ 𝐴 ∈ V)
32	30, 31	impbii 209	1 ⊢ (TC+ 𝐴 ∈ V ↔ TC+ 𝐴 ∈ ∪ (𝑅1 “ On))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429   ∖ cdif 3886   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273  ∪ cuni 4850   “ cima 5634  Oncon0 6323  𝑅₁cr1 9686  TC+ cttc 36668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-r1 9688  df-ttc 36669

This theorem is referenced by:  ttcwf3  36708